Multiscale analysis and simulation of a signalling process with surface diffusion

TL;DR

This paper develops a multiscale model for cell signalling involving surface diffusion and reactions, proving convergence to a macroscopic limit and introducing a finite element method for simulation.

Contribution

It introduces a two-scale convergence analysis and a finite element scheme for a complex cell signalling model with surface diffusion.

Findings

Convergence of the model solutions to a two-scale macroscopic problem.

Development of a finite element method for the multiscale model.

Numerical simulations demonstrating the impact of cell heterogeneities.

Abstract

We present and analyse a model for cell signalling processes in biological tissues. The model includes diffusion and nonlinear reactions on the cell surfaces, and both inter- and intracellular signalling. Using techniques from the theory of two-scale convergence as well the unfolding method, we show convergence of the solutions to the model to solutions of a two-scale macroscopic problem. We also present a two-scale bulk-surface finite element method for the approximation of the macroscopic model. We report on some benchmarking results as well as numerical simulations in a biologically relevant regime that illustrate the influence of cell-scale heterogeneities on macroscopic concentrations.